Looks like fowl pox. Yes, the others will get it. Looks like the dry type, not a huge deal. If you want to, you can dab the spots with iodine to dry them up or leave them alone and they will on their own.
Some of the viruses that chickens contract are herpes-type viruses, meaning they become carriers of that disease even after they seem to recover.
I think it's okay to eat the eggs; after all, you aren't medicating them with antibiotics, so why not?